Break Even Point

Question

Suppose that x thousand units of a product will be sold when the price is p(x) = 50 - 1.25x dollars per unit and the cost of producing x thousand units is C(x) = 20x + 100 thousand dollars.

a) What levels of production result in the company just breaking even?

Would this be setting p(x) = C(x)?

b) What price per unit would would result in the company just breaking even?

Answer 1

If "breaking even" means you spend as much as you receive, then you need to set those quantities equal.

There are (at least) two ways you could measure this:

cost per item = price per item

total cost = total sales

In your problem, $p(x)$ is the price per item, but $C(x)$ is the total cost, so you're comparing apples to oranges: note that looking at the units tells you this, since "dollars per item" and "thousands of dollars" are different kinds of units.

Instead, you're going to either have to compute total sales, or cost per item.

(strictly speaking, breaking even means "total cost = total sales", but it's not hard to see that if "cost per item = price per item" that you must also have "total cost = total sales")

(also, take care that, if you get in a situation where you're comparing "dollars" to "thousands of dollars" that you do an appropriate unit conversion)

Answer 2

The break-even point (BEP) in terms of (Thousand) Unit Sales $x$ can be directly computed in terms of Total Revenue $r(x)=x\times p(x)$ (price $\times$ quantity) and Total Costs $c(x)$ as the quantity at which the profit $\pi(x)=r(x)-c(x)=0$, that is $$ p(x)x-c(x) = (50 - 1.25x)x-(20x + 100)=0\quad\Longrightarrow \quad x_1=4,\,x_2=20 $$ so the profit will be positive if the production is beetwen $x_1$ and $x_2$: $$ \pi(x)\ge 0 \quad\text{for }x_1\le x\le x_2 $$ So the prices at the break-even points are $p_1=p(x_1)=45$ and $p_2=p(x_2)=25$.